A343379 Number of strict integer partitions of n with no part dividing or divisible by all the other parts.

1, 0, 0, 0, 0, 1, 0, 2, 1, 3, 3, 5, 3, 9, 9, 12, 12, 18, 18, 27, 27, 36, 41, 51, 51, 73, 80, 96, 105, 132, 137, 177, 188, 230, 253, 303, 320, 398, 431, 508, 550, 659, 705, 847, 913, 1063, 1165, 1359, 1452, 1716, 1856, 2134, 2329, 2688, 2894, 3345, 3622, 4133

Offset

0,8

Comments

Alternative name: Number of strict integer partitions of n that are either empty, or (1) have smallest part not dividing all the others and (2) have greatest part not divisible by all the others.

Links

Table of n, a(n) for n=0..57.

Formula

The Heinz numbers for the non-strict version are A343338 = A342193 /\ A343337.

Example

The a(5) = 1 through a(13) = 9 partitions (empty column indicated by dot):
  (3,2)  .  (4,3)  (5,3)  (5,4)    (6,4)    (6,5)    (7,5)    (7,6)
            (5,2)         (7,2)    (7,3)    (7,4)    (5,4,3)  (8,5)
                          (4,3,2)  (5,3,2)  (8,3)    (7,3,2)  (9,4)
                                            (9,2)             (10,3)
                                            (5,4,2)           (11,2)
                                                              (6,4,3)
                                                              (6,5,2)
                                                              (7,4,2)
                                                              (8,3,2)

Mathematica

Table[Length[Select[IntegerPartitions[n], #=={}||UnsameQ@@#&&!And@@IntegerQ/@(#/Min@@#)&&!And@@IntegerQ/@(Max@@#/#)&]], {n, 0, 30}]

Crossrefs

The first condition alone gives A341450.

The non-strict version is A343342 (Heinz numbers: A343338).

The second condition alone gives A343377.

The opposite version is A343378.

The half-opposite versions are A343380 and A343381.

The version for "or" instead of "and" is A343382.

A000009 counts strict partitions.

A000070 counts partitions with a selected part.

A006128 counts partitions with a selected position.

A015723 counts strict partitions with a selected part.

A018818 counts partitions into divisors (strict: A033630).

A167865 counts strict chains of divisors > 1 summing to n.

A339564 counts factorizations with a selected factor.

Cf. A083710, A097986, A200745, A264401, A338470, A339562, A342193, A343337, A343341, A343343, A343346, A343347.

Sequence in context: A355659 A152993 A178133 * A026927 A240863 A288005

Adjacent sequences: A343376 A343377 A343378 * A343380 A343381 A343382

Keyword

nonn

Author

Gus Wiseman, Apr 16 2021

Status

approved